The Stacks project

Definition 71.18.1. With $Z \subset B$ and $f : X \to B$ as above.

  1. Given a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the strict transform of $\mathcal{F}$ with respect to the blowup of $B$ in $Z$ is the quotient $\mathcal{F}'$ of $\text{pr}_ X^*\mathcal{F}$ by the submodule of sections supported on $|\text{pr}_{B'}^{-1}E|$.

  2. The strict transform of $X$ is the closed subspace $X' \subset X \times _ B B'$ cut out by the quasi-coherent ideal of sections of $\mathcal{O}_{X \times _ B B'}$ supported on $|\text{pr}_{B'}^{-1}E|$.

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